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Claim. Write $H_n=1+1/2+\cdots+1/n=u_n/v_n$ in lowest terms, let $\mathcal L$ be the set of positive integers $n$ with $v_n$ less than $\mathrm{lcm}(1,\ldots,n)$, and let $\bar d(\mathcal L)$ be its upper asymptotic density. The paper's Theorem 2 (p. 54) states: "Assuming Conjecture 1, we have $\bar{d}(\mathcal{L})=1$." In the notation of Problem 291, $\sum_{k\le n}1/k=a_n/L_n$ gives $v_n=L_n/(a_n,L_n)$, so $n\in\mathcal L$ exactly when $(a_n,L_n)>1$. Under the hypothesis the set of $n$ with $(a_n,L_n)>1$ therefore has upper density $1$, and in particular is infinite: this is the second question, which three partial pages answer unconditionally. Upper density $1$ does not exclude infinitely many $n$ with $(a_n,L_n)=1$, so nothing follows for the first question.

Hypothesis. The paper's Conjecture 1 (p. 54): for any distinct primes $q_1,\ldots,q_l$, the numbers $1/\log q_1,\ldots,1/\log q_l$ are linearly independent over $\mathbb Q$. The paper derives it from the weak Schanuel conjecture: for nonzero, multiplicatively independent algebraic numbers $\beta_1,\ldots,\beta_m$ the numbers $\log\beta_1,\ldots,\log\beta_m$ are algebraically independent; applied to distinct primes, this makes their logarithms, and so the reciprocals of the logarithms, algebraically independent. Both conjectures are open, so this page derives nothing for the problem's standing. The proof uses the hypothesis through Kronecker's theorem, applied to the numbers $\log p_2/\log p_i$, together with Mertens' theorem.

Depends on. Nothing in this wiki; the claim rests on the cited paper and on Conjecture 1.

Standing. Refereed: C. R. Math. Acad. Sci. Paris 360 (2022), 53--57 (received 11 August 2021, accepted 12 October 2021, published online 26 January 2022), with no arXiv version. The site's curator, Thomas Bloom, reports the theorem in the problem's commentary, but the site labels the problem OPEN, so that report is not acceptance and reviewed is not listed. The formal-conjectures variant erdos_291.variants.wu_yan states the theorem with the independence hypothesis as an explicit argument and a sorry body; it is a statement, not a formalization, and gives no formalized evidence. The library's source card records the statements of Conjecture 1 and Theorem 2 checked clause by clause and the two-page proof read through, not verified.